Kelly swallowed hard, a top aide recounted: For New York, it was “good news with complications.”

Those “complications” included the emergency security that the cash-strapped police department would mobilize to protect New York’s 8.5 million people against a possible revenge or retaliatory attack by militant Islamists.

It was a great day for America, Kelly told his top aides — especially for New Yorkers, who for a decade have been constantly reminded of that terrible September day by the vast pit where the Twin Towers once stood.

Yet bin Laden’s death has significantly increased the risk of more terrorism directed at New York, militant Islam’s No. 1 target.

Kelly immediately called Mayor Bloomberg, who was himself being briefed about the impending announcement by Homeland Security chief Janet Napolitano.

By the time the president was telling the nation of the stunning raid, the NYPD had a major expansion of security underway. A “FINEST” message instructing cops to be prepared in light of the impending announcement had already gone out to all commands. The city’s 76 precincts were ordered to increase security around station houses and take other precautions, such as paying even closer attention to suspicious packages and people.

Kelly also directed that a midnight tour of cops working transit hubs be held over, almost doubling the number of police deployed in subways and around the city’s train and ferry stations. As many as 100 “Critical Response Vehicles” holding 200 cops were ordered to rally at a designated center — a practice conducted at least twice a day somewhere in the city.

The next morning, New Yorkers emerging from rush-hour commutes in subway and train stations encountered extra police and explosive-sniffing dogs and bag-check posts throughout the city. The NYPD ordered a similar increase for the evening rush hour.

That increased presence is still in force, subject to day-to-day evaluation. While Napolitano and city officials have stressed that they have gotten no specific threat warnings that would boost the new alert system to a higher level, the NYPD is taking no chances.

The added presence of law enforcement in city streets and at iconic structures like the Empire State Building and Grand Central Station is likely to continue at least through tomorrow, when Obama is to visit the city for a ceremony at the World Trade Center site.

New Yorkers seem accustomed by now to the extra police presence and added security measures. Many aren’t even aware of the changes that the 9/11 strikes have brought to the city. Did you know that each of the city’s precincts now dedicates at least one patrol car to routine checks on houses of worship and other sensitive sites?

Kelly now allocates some $330 million a year of his $4.6 billion budget to counterterrorism, along with over a 1,000 of his 50,000-person force — no mean feat in a department whose uniformed ranks are now below 37,000, down about 10 percent from the pre-9/11 level.

The NYPD is also still fighting for federal funds to finance 70 percent of Kelly’s ambitious scheme to install sophisticated surveillance cameras and license-plate readers in Manhattan — a $500 million system that he calls vital to the city’s security.

Osama is rotting beneath the sea, but the threat to New York is likely to grow in the coming months as jihadis try to avenge their hero’s murder. The NYPD doesn’t get to uncork the champagne.